Abstract
An extended Green's function method is presented for a localized antiferromagnetic (AF) system with mixed AF and ferromagnetic couplings, and single-ion anisotropy. Expressions are given for the N\'eel temperature, the temperature-dependent magnetic order parameter (MOP), and the spin component along the preferred direction of the single-ion anisotropy in the ground state for $A$-type AF $\mathrm{La}\mathrm{Mn}{\mathrm{O}}_{3}$. These predict the MOP found by inelastic neutron scattering. Our theory exhibits a nontrivial internal consistency with four measured parameters, including ${T}_{N}$. Our theory also predicts a temperature-dependent spin energy gap that is different from the Holstein-Primakoff theory.
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